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arXiv:cond-mat/9606218 · TURKISH JOURNAL OF PHYSICS

Pairing Mechanism in two Hubbard Models That Show d_{x^2-y^2} Pairing

Douglas J. Scalapino

Published Jan 1, 1996• 3 citations
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Abstract

Here we examine various aspects of the pairing mechanism for two models, the two-dimensional and two-leg ladder Hubbard models. Both of these models exhibit pairing correlations with d_{x^{2}-y^{2}} symmetry. However, the undoped insulating states of these two systems are different with the two-dimensional lattice characterized by a ground state with long-range antiferromagnetic order and the two-leg ladder having a spin gap in its ground state. Our aim is to gain a better understanding of the underlying pairing mechanism which causes d_{x^{2}-y^{2}} pairing in these two models.

DOI
10.55730/1300-0101.2544

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Here we examine various aspects of the pairing mechanism for two models, the two-dimensional and two-leg ladder Hubbard models. Both of these models exhibit pairing correlations with d_{x^{2}-y^{2}} symmetry. However, the undoped insulating states of these two systems are different with the two-dimensional lattice characterized by a ground state with long-range antiferromagnetic order and the two-leg ladder having a spin gap in its ground state. Our aim is to gain a better understanding of the underlying pairing mechanism which causes d_{x^{2}-y^{2}} pairing in these two models.
Asymmetric two-leg Hubbard ladders with different on-site interactions Uy{U}_{y}Uy​ and hoppings ty{t}_{y}ty​ on each leg are investigated using the density-matrix renormalization group method and exact diagonalizations. The pairing found in symmetric ladders is robust against the introduction of the leg asymmetry. When studying pairing, one-band Hubbard ladder models are better described as one-dimensional correlated two-band models than as sublattices of higher-dimensional systems. The asymmetric Hubbard ladder provides us with a simple model for studying pairing in the crossover regime between charge-transfer and Mott insulators.
Short-range antiferromagnetic correlations are known to open a spin gap in the repulsive Hubbard model on ladders with \boldsymbol{M} 𝐌 legs, when \boldsymbol{M} 𝐌 is even. We show that the spin gap originates from the formation of correlated pairs of electrons with opposite spin, captured by the hidden ordering of a spin-parity operator. Since both spin gap and parity vanish in the two-dimensional limit, we introduce the fractional generalization of spin parity and prove that it remains finite in the thermodynamic limit. Our results are based upon variational wave functions and Monte Carlo calculations: performing a finite size-scaling analysis with growing \boldsymbol{M} 𝐌 , we show that the doping region where the parity is finite coincides with the range in which superconductivity is observed in two spatial dimensions. Our observations support the idea that superconductivity emerges out of spin gapped phases on ladders, driven by a spin-pairing mechanism, in which the ordering is conveniently captured by the finiteness of the fractional spin-parity operator.
In order to shed light whether the `even-odd conjecture' (even numbers of legs will superconduct accompanied by a spin gap while odd ones do not) for correlated electrons in ladder systems, the pairing correlation is studied for the Hubbard model on a two- and three-leg ladders. We have employed both the weak-coupling renormalization group and the quantum Monte Carlo (QMC) method for strong interactions. For the two-leg Hubbard ladder, a systematic QMC (with a controlled level spacings) has detected an enhanced pairing correlation, which is consistent with the weak-coupling prediction. We also calculate the correlation functions in the three-leg Hubbard ladder and show that the weak-coupling study predicts the dominant superconductivity, which refutes the naive even-odd conjecture. A crucial point is a spin gap for only some of the multiple spin modes is enough to make the ladder superconduct with a pairing symmetry (d-like here) compatible with the gapped mode. A QMC study for the three-leg ladder endorses the enhanced pairing correlation.
Using the Density Matrix Renormalization Group and two-leg ladders, we investigate an electronic two-orbital Hubbard model including plaquette-diagonal hopping amplitudes. Our goal is to search for regimes where charges added to the undoped state form pairs, presumably a precursor of a superconducting state. For the electronic density $\ensuremath{\rho}=2$, i.e., the undoped limit, our investigations show a robust $(\ensuremath{\pi},0)$ antiferromagnetic ground state, as in previous investigations. Doping away from $\ensuremath{\rho}=2$ and for large values of the Hund coupling JJJ, a ferromagnetic region is found to be stable. Moreover, when the interorbital on-site Hubbard repulsion is smaller than the Hund coupling, i.e., for ${U}^{\ensuremath{'}}<J$ in the standard notation of multiorbital Hubbard models, our results indicate the coexistence of pairing tendencies and ferromagnetism close to $\ensuremath{\rho}=2$. These results are compatible with previous investigations using one-dimensional systems. Although further research is needed to clarify if the range of couplings used here is of relevance for real materials, such as superconducting heavy fermions or pnictides, our theoretical results address a possible mechanism for pairing that may be active in the presence of short-range ferromagnetic fluctuations.
Using density matrix renormalization group calculations, we compare results obtained for the $t\ensuremath{-}J,$ one-band Hubbard, and three-band Hubbard models of a two-leg CuO ladder. Spin and charge gaps, pair binding energies, and effective pair hoppings are calculated for a wide range of parameters. All three models have an insulating state at a filling corresponding to one hole per Cu site. For physically relevant parameters their spin gaps are similar in size but they exhibit quite different charge gaps. We find that the binding energy of a pair of doped holes is of the order of the undoped ladder spin gap for all three models. The main difference between the models is the size of the effective pair hopping, which is significantly larger in the three-band model with parameters appropriate for CuO materials than in the other two models.
It is shown, using asymptotically exact methods, that the two-dimensional repulsive Hubbard model with strongly modulated interactions exhibits ``high-temperature superconductivity.'' Specifically, the explicit modulation, which has the same symmetry as period 4 bond-centered stripes, breaks the system into an alternating array of more and less heavily hole-doped, nearly decoupled two-leg ladders. It is shown that this system exhibits a pairing scale determined by the spin gap of the undoped two-leg ladder, and a phase ordering temperature proportional to a low positive power of the interladder coupling.